Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 3.2.1.6. Let $(X,x)$ be a pointed simplicial set and let $(Y,y)$ be a pointed topological space. Suppose we are given a pair of continuous functions $f_0, f_1: | X_{} | \rightarrow Y$ carrying $x$ to $y$, which we can identify with pointed morphisms $f'_0, f'_1: X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$. Let $h: [0,1] \times | X_{} | \rightarrow Y$ be a continuous function satisfying $f_0 = h|_{ \{ 0\} \times |X_{}|}$, $f_1 = h|_{ \{ 1\} \times | X_{} |}$, and $h(t,x) = y$ for $0 \leq t \leq 1$ (that is, $h$ is a pointed homotopy from $f_0$ to $f_1$ in the category of topological spaces). Then the composite map

\[ | \Delta ^1 \times X_{} | \xrightarrow {\theta } | \Delta ^1 | \times | X_{} | = [0,1] \times | X_{} | \xrightarrow {h} Y \]

classifies a morphism of simplicial sets $h': \Delta ^1 \times X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$, which is a pointed homotopy from $f'_0$ to $f'_1$ (in the sense of Definition 3.2.1.4). By virtue of Theorem , the map $\theta $ is a homeomorphism, so every pointed homotopy from $f_0$ to $f_1$ arises in this way. In other words, the construction $h \mapsto h'$ induces a bijection

\[ \xymatrix { \{ \text{(Continuous) pointed homotopies from $f_0$ to $f_1$} \} \ar [d]^{\sim } \\ \{ \text{(Simplicial) pointed homotopies from $f'_0$ to $f'_1$} \} . } \]